Light Transport Jet Captain reported during initial climb from OZW the radio altimeter and EGPWS began to malfunction; resulting in distraction and an altitude deviation.

Narrative

Moments after liftoff and gear retraction we started getting 'Terrain' alerts along with other GPWS callouts. We were climbing in a left turn to a 170 degree heading and 3;000 feet as assigned by ATC. During the maneuver the pilot flying shallowed the bank as we analyzed the situation. The GPWS system continued to make callouts such as 'Terrain' 'Pull Up' 'Gear - Gear' and during the distraction the pilot flying overshot our assigned altitude by 400 feet then undershot by 300 or so feet till leveling at 3;000 as we analyzed the misleading callouts.I took over flying the aircraft and talked to ATC as we troubleshot the problem. ATC cleared us to 4;000 feet and we kept our airspeed low and continued under radar vectors to stay in their airspace as we continued to get alerts such as 'Too Low' and 'Gear' intermittently.After a few minutes the alerts stopped; and we advised ATC we were able to continue on course.After discussion; we thought it to be erroneous signals from the radar altimeter.There were no conflicts with other aircraft and the Detroit controller did a great job in assisting our requests.